The field of definition for dynamical systems on P^N

Abstract

Let Hom^N_d be the set of morphisms of degree d from P^N to itself. For f an element of PGL_{N+1}, let phi^f represent the conjugation action f^{-1} phi f. Let M^N_d = Hom_d^N/PGL_{N+1} be the moduli space of degree d morphisms of P^N. A field of definition for class of morphisms is a field over which at least one morphism in the class is defined. The field of moduli for a class of morphisms is the fixed field of the set of Galois elements fixing that class. Every field of definition contains the field of moduli. In this article, we give a sufficient condition for the field of moduli to be a field of definition for morphisms whose stabilizer group is trivial.